No, the counseling/registration for JoSAA 2019 has not started yet. JEE ADV results are going to be declared today, and in the third week of June, that is, about next week the JoSAA counseling is going to start. So to apply for that, you just have to go to the JoSAA official website, and Sign in through your JEE MAINS application number. The latter process is fairly simple. You just have to set a preference list according to the course and the college that you want, and you will be allotted the college according to your merit and rank. All the best!